This activity involves rounding four-digit numbers to the nearest thousand.

Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?

Find a route from the outside to the inside of this square, stepping on as many tiles as possible.

What happens when you round these numbers to the nearest whole number?

In this problem we are looking at sets of parallel sticks that cross each other. What is the least number of crossings you can make? And the greatest?

What happens when you round these three-digit numbers to the nearest 100?

Here are two kinds of spirals for you to explore. What do you notice?

In this game for two players, the idea is to take it in turns to choose 1, 3, 5 or 7. The winner is the first to make the total 37.

Can you explain the strategy for winning this game with any target?

Use two dice to generate two numbers with one decimal place. What happens when you round these numbers to the nearest whole number?

Delight your friends with this cunning trick! Can you explain how it works?

Nim-7 game for an adult and child. Who will be the one to take the last counter?

Are these statements relating to odd and even numbers always true, sometimes true or never true?

Only one side of a two-slice toaster is working. What is the quickest way to toast both sides of three slices of bread?

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

Four bags contain a large number of 1s, 3s, 5s and 7s. Pick any ten numbers from the bags above so that their total is 37.

This challenge focuses on finding the sum and difference of pairs of two-digit numbers.

Can you find an efficient method to work out how many handshakes there would be if hundreds of people met?

Investigate the sum of the numbers on the top and bottom faces of a line of three dice. What do you notice?

This task follows on from Build it Up and takes the ideas into three dimensions!

How could Penny, Tom and Matthew work out how many chocolates there are in different sized boxes?

Can you find all the ways to get 15 at the top of this triangle of numbers?

This challenge, written for the Young Mathematicians' Award, invites you to explore 'centred squares'.

Find some examples of pairs of numbers such that their sum is a factor of their product. eg. 4 + 12 = 16 and 4 × 12 = 48 and 16 is a factor of 48.

Triangular numbers can be represented by a triangular array of squares. What do you notice about the sum of identical triangle numbers?

One block is needed to make an up-and-down staircase, with one step up and one step down. How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

If you can copy a network without lifting your pen off the paper and without drawing any line twice, then it is traversable. Decide which of these diagrams are traversable.

List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?

Sweets are given out to party-goers in a particular way. Investigate the total number of sweets received by people sitting in different positions.

A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?

Imagine starting with one yellow cube and covering it all over with a single layer of red cubes, and then covering that cube with a layer of blue cubes. How many red and blue cubes would you need?

Try entering different sets of numbers in the number pyramids. How does the total at the top change?

Use your addition and subtraction skills, combined with some strategic thinking, to beat your partner at this game.

You can work out the number someone else is thinking of as follows. Ask a friend to think of any natural number less than 100. Then ask them to tell you the remainders when this number is divided by. . . .

The number of plants in Mr McGregor's magic potting shed increases overnight. He'd like to put the same number of plants in each of his gardens, planting one garden each day. How can he do it?

Watch this video to see how to roll the dice. Now it's your turn! What do you notice about the dice numbers you have recorded?

What would be the smallest number of moves needed to move a Knight from a chess set from one corner to the opposite corner of a 99 by 99 square board?

Can you work out how to win this game of Nim? Does it matter if you go first or second?

Find the sum and difference between a pair of two-digit numbers. Now find the sum and difference between the sum and difference! What happens?

The NRICH team are always looking for new ways to engage teachers and pupils in problem solving. Here we explain the thinking behind maths trails.

How many ways can you find to do up all four buttons on my coat? How about if I had five buttons? Six ...?

Got It game for an adult and child. How can you play so that you know you will always win?

It would be nice to have a strategy for disentangling any tangled ropes...

When number pyramids have a sequence on the bottom layer, some interesting patterns emerge...

Place the numbers from 1 to 9 in the squares below so that the difference between joined squares is odd. How many different ways can you do this?

Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?

Are these statements always true, sometimes true or never true?

A game for two people, or play online. Given a target number, say 23, and a range of numbers to choose from, say 1-4, players take it in turns to add to the running total to hit their target.